<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns:mv="http://macVmlSchemaUri" xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta name=Title content="Front propagation in Hele-Shaw cells"> <meta name=Keywords content=""> <meta http-equiv=Content-Type content="text/html; charset=unicode"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 14"> <meta name=Originator content="Microsoft Word 14"> <link rel=File-List href="HeleShawCells_files/filelist.xml"> <link rel=Edit-Time-Data href="HeleShawCells_files/editdata.mso"> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--> <title>Front propagation in Hele-Shaw cells</title> <!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Author>Paul Ronney</o:Author> <o:Template>Normal</o:Template> <o:LastAuthor>Paul Ronney</o:LastAuthor> <o:Revision>8</o:Revision> <o:TotalTime>7</o:TotalTime> <o:Created>2002-07-01T15:51:00Z</o:Created> <o:LastSaved>2015-02-11T22:50:00Z</o:LastSaved> <o:Pages>1</o:Pages> <o:Words>816</o:Words> <o:Characters>4656</o:Characters> <o:Company>Future Soccer Stars of America</o:Company> <o:Lines>38</o:Lines> <o:Paragraphs>10</o:Paragraphs> <o:CharactersWithSpaces>5462</o:CharactersWithSpaces> <o:Version>14.0</o:Version> </o:DocumentProperties> <o:OfficeDocumentSettings> <o:AllowPNG/> </o:OfficeDocumentSettings> </xml><![endif]--> <link rel=themeData href="HeleShawCells_files/themedata.xml"> <!--[if gte mso 9]><xml> <w:WordDocument> <w:Zoom>125</w:Zoom> <w:SpellingState>Clean</w:SpellingState> <w:GrammarState>Clean</w:GrammarState> <w:TrackMoves>false</w:TrackMoves> <w:TrackFormatting/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-US</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:SplitPgBreakAndParaMark/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="&#45;-"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true" DefSemiHidden="true" DefQFormat="false" DefPriority="99" LatentStyleCount="276"> <w:LsdException Locked="false" Priority="0" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" Priority="9" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" Priority="39" Name="toc 1"/> <w:LsdException Locked="false" Priority="39" Name="toc 2"/> <w:LsdException Locked="false" Priority="39" Name="toc 3"/> <w:LsdException Locked="false" Priority="39" Name="toc 4"/> <w:LsdException Locked="false" Priority="39" Name="toc 5"/> <w:LsdException Locked="false" Priority="39" Name="toc 6"/> <w:LsdException Locked="false" Priority="39" Name="toc 7"/> <w:LsdException Locked="false" Priority="39" Name="toc 8"/> <w:LsdException Locked="false" Priority="39" Name="toc 9"/> <w:LsdException Locked="false" Priority="35" QFormat="true" Name="caption"/> <w:LsdException Locked="false" Priority="10" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Title"/> <w:LsdException Locked="false" Priority="1" Name="Default Paragraph Font"/> <w:LsdException Locked="false" Priority="11" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" Priority="22" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" Priority="20" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" Priority="59" SemiHidden="false" UnhideWhenUsed="false" Name="Table Grid"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Placeholder Text"/> <w:LsdException Locked="false" Priority="1" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="No Spacing"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 1"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 1"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 1"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 1"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 1"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Revision"/> <w:LsdException Locked="false" Priority="34" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="List Paragraph"/> <w:LsdException Locked="false" Priority="29" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Quote"/> <w:LsdException Locked="false" Priority="30" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Quote"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 1"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 1"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 1"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 1"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 1"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 1"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 1"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 2"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 2"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 2"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 2"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 2"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 2"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 2"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 2"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 2"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 2"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 2"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 3"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 3"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 3"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 3"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 3"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 3"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 3"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 3"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 3"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 3"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 3"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 3"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 3"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 4"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 4"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 4"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 4"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 4"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 4"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 4"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 4"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 4"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 4"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 4"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 4"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 4"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 4"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 5"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 5"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 5"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 5"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 5"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 5"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 5"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 5"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 5"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 5"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 5"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 5"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 5"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 5"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 6"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 6"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 6"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 6"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 6"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 6"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles> </xml><![endif]--> <style> <!-- /* Font Definitions */ @font-face {font-family:Arial; panose-1:2 11 6 4 2 2 2 2 2 4; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:10887 -2147483648 8 0 511 0;} @font-face {font-family:Times; panose-1:2 0 5 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:"-3 fg"; mso-font-charset:78; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-536870145 1791491579 18 0 131231 0;} @font-face {font-family:"-3 fg"; mso-font-charset:78; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-536870145 1791491579 18 0 131231 0;} @font-face {font-family:"Lucida Grande"; panose-1:2 11 6 0 4 5 2 2 2 4; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-520090897 1342218751 0 0 447 0;} @font-face {font-family:"\@-3 fg"; mso-font-charset:78; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-536870145 1791491579 18 0 131231 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:10.0pt; font-family:Times; mso-fareast-font-family:"-3 fg"; mso-fareast-theme-font:minor-fareast; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} a:link, span.MsoHyperlink {mso-style-noshow:yes; mso-style-priority:99; color:blue; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {mso-style-noshow:yes; mso-style-priority:99; color:blue; text-decoration:underline; text-underline:single;} p {mso-style-noshow:yes; mso-style-priority:99; mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:10.0pt; font-family:Times; mso-fareast-font-family:"-3 fg"; mso-fareast-theme-font:minor-fareast; mso-bidi-font-family:"Times New Roman";} p.MsoAcetate, li.MsoAcetate, div.MsoAcetate {mso-style-noshow:yes; mso-style-priority:99; mso-style-link:"Balloon Text Char"; mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:9.0pt; font-family:"Lucida Grande"; mso-fareast-font-family:"-3 fg"; mso-fareast-theme-font:minor-fareast;} span.BalloonTextChar {mso-style-name:"Balloon Text Char"; mso-style-noshow:yes; mso-style-priority:99; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Balloon Text"; mso-ansi-font-size:9.0pt; mso-bidi-font-size:9.0pt; font-family:"Lucida Grande"; mso-ascii-font-family:"Lucida Grande"; mso-fareast-font-family:"-3 fg"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:"Lucida Grande"; mso-bidi-font-family:"Lucida Grande";} span.SpellE {mso-style-name:""; mso-spl-e:yes;} span.GramE {mso-style-name:""; mso-gram-e:yes;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt;} @page WordSection1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} --> </style> <!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--><!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="1026"> <v:textbox inset="5.85pt,.7pt,5.85pt,.7pt"/> </o:shapedefaults></xml><![endif]--><!--[if gte mso 9]><xml> <o:shapelayout v:ext="edit"> <o:idmap v:ext="edit" data="1"/> </o:shapelayout></xml><![endif]--> </head> <body lang=EN-US link=blue vlink=blue style='tab-interval:.5in' bgproperties=fixed> <div class=WordSection1> <p align=center style='text-align:center'><strong><span style='font-size:14.0pt; font-family:Arial;mso-bidi-theme-font:major-latin;layout-grid-mode:line'>Chemical front and flame front propagation in <span class=SpellE>Hele</span>-Shaw cells</span></strong><span style='font-size:14.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><o:p></o:p></span></p> <p><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><a href="http://ronney.usc.edu/HeleShaw/">View on-line presentation on aqueous chemical fronts in Hele-Shaw cells</a><span style='color:fuchsia'><br> </span><a href="http://ronney.usc.edu/HeleShaw/HeleShaw.html">View on-line presentation on gaseous flames in Hele-Shaw cells</a><span style='color:fuchsia'><br> </span><a href="http://ronney.usc.edu/HeleShaw/heleshaw.mov"><span style='color:fuchsia'>Download movie on gaseous flames in Hele-Shaw cells (48 MB!)</span></a><o:p></o:p></span></p> <p><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'>It is well known that buoyancy and thermal expansion affect the propagation rates and shapes of premixed gas flames. The understanding of such effects is complicated by the large density ratio between the reactants and products, which induces a <span class=SpellE>baroclinic</span> production of <span class=SpellE>vorticity</span> due to misalignment of density and pressure gradients at the front, which in turn leads to a complicated multi-dimensional flame/flow interaction. The <span class=SpellE>Hele</span>-Shaw cell,<i> i.e</i>., the region between <span class=GramE>closely-spaced</span> flat parallel plates, is probably the simplest system in which multi-dimensional convection is present, consequently, the behavior of fluids in this system has been studied extensively (<span class=SpellE>Homsy</span>, 1987). Probably the most important characteristic of <span class=SpellE>Hele</span>-Shaw flows is that when the Reynolds number based on gap width is sufficiently small, the <span class=SpellE>Navier</span>-Stokes equations averaged over the gap reduce to a <u>linear</u> relation, namely a Laplace equation for pressure (Darcy's law). <o:p></o:p></span></p> <p><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'>In this work, flame propagation in <span class=SpellE>Hele</span>-Shaw cells is studied to obtain a better understanding of buoyancy and thermal expansion effects on premixed flames. This work is also relevant to the study of unburned hydrocarbon emissions produced by internal combustion engines since these emissions are largely a result of the partial burning or complete flame quenching in the narrow, annular gap called the &quot;crevice volume&quot; between the piston and cylinder walls (see, for example, the text by J. B. Heywood, 1988). A better understanding of how flames propagate in these volumes through experiments using <span class=SpellE>Hele</span>-Shaw cells could lead to identification of means to reduce these emissions.<o:p></o:p></span></p> <p><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'>Because of the very weak thermal expansion (typically 0.06%) caused by chemical reaction, the aqueous chemical fronts are affected by buoyancy. This phenomenon has been studied in <span class=SpellE>Hele</span>-Shaw cells, <em><span style='font-family:Arial;mso-bidi-theme-font:major-latin'>i.e</span></em>., the gap between two closely spaced flat parallel plates. Results show a new type of fingering mechanism not present in non-reacting <span class=SpellE>Hele</span>-Shaw flows, which has been identified as a surface tension effect, even though the reactant and product solutions are miscible in all proportions. In fact, this wavelength is almost independent of the front propagation speed (S or S<sub>L</sub>) and the cell thickness (w). The only viable explanation of this, when compared to the predictions of the <span class=SpellE>Saffman</span>-Taylor model, is a surface tension at the interface whose magnitude is about 0.005 dyne/cm  about <u>14,000 times</u> smaller than that of a water-air interface.<o:p></o:p></span></p> <div align=center> <table class=MsoNormalTable border=1 cellpadding=0 style='mso-cellspacing:1.5pt; border:outset 1.5pt;mso-yfti-tbllook:1184;mso-padding-alt:0in 0in 0in 0in'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal style='margin:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1032" type="#_x0000_t75" style='width:225pt;height:84pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image001.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=227 height=86 src="HeleShawCells_files/image002.gif" v:shapes="Picture_x0020_1"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><br> </span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font: major-latin;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1031" type="#_x0000_t75" style='width:225pt;height:83pt; visibility:visible;mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image003.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=227 height=85 src="HeleShawCells_files/image004.gif" v:shapes="Picture_x0020_2"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><o:p></o:p></span></p> </td> <td style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal style='margin:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_3" o:spid="_x0000_i1030" type="#_x0000_t75" style='width:385pt;height:269pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image005.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=387 height=271 src="HeleShawCells_files/image006.gif" v:shapes="Picture_x0020_3"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><o:p></o:p></span></p> </td> </tr> </table> </div> <p style='margin-top:5.0pt;margin-right:.5in;margin-bottom:5.0pt;margin-left: .5in'><em><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font: major-latin;layout-grid-mode:line'>Left: images of upward propagating autocatalytic fronts in a <span class=SpellE>Hele</span>-Shaw cell, cell thickness (w) = 1.0 mm, S<sub>L</sub> = 0.17 mm/s. Upper image: 10 seconds after initiation<span class=GramE>;</span> lower image: after reaching quasi-steady propagation condition. Width of cell is 200 mm. Right: effect of w and S<sub>L</sub> (<span class=SpellE>Peclet</span> number = <span class=SpellE>Sw</span>/D, where D = mass diffusivity of the <span class=SpellE>stoichiometrically</span> limiting reactant, IO<sub>3</sub><sup>-</sup>) on wavelength of initial disturbance.</span></em><span style='font-size:12.0pt;font-family:Arial; mso-bidi-theme-font:major-latin;layout-grid-mode:line'><o:p></o:p></span></p> <p><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'>Remarkably, the propagation rates of these wrinkled buoyant fronts also conform to <span class=SpellE>Yakhot's</span> predictions when a characteristic linear growth rate of the buoyancy-induced instability is used to estimate the effective turbulence intensity (see second plot on this page). <o:p></o:p></span></p> <p><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'>As a complement to the experiments on chemical fronts in <span class=SpellE>Hele</span>-Shaw cells, premixed-gas flames in <span class=SpellE>Hele</span>-Shaw cells were also examined. Significantly, wrinkling was observed even for downward propagating (buoyantly stable) flames and flames having high Lewis number (diffusive-thermally stable). The burning rates (S<sub>T</sub>) of these flames are quite different from their laminar, unwrinkled values (S<sub>L</sub>). Values of S<sub>T</sub>/S<sub>L</sub> in the quasi-steady stage were higher for upward vs. downward propagation, but only weakly dependent on Lewis and <span class=SpellE>Peclet</span> number. Due to these wrinkling effects, the front propagation rates in <span class=SpellE>Hele</span>-Shaw cells are found to be <strong><span style='font-family:Arial;mso-bidi-theme-font: major-latin'>always</span></strong> faster than the laminar flame speed, typically by a factor of 3.These results show that <em><span style='font-family: Arial;mso-bidi-theme-font:major-latin'>even for diffusively stable mixtures, at microgravity thermal expansion and viscosity changes across the front will lead to flame instabilities</span></em>. <strong><span style='font-family:Arial; mso-bidi-theme-font:major-latin'>These results also indicate that the behavior of flame propagation in narrow channels such as crevice volumes in premixed-charge internal combustion engines (the source of most unburned hydrocarbon emissions) may be quite different from that inferred from simple laminar flame experiments.</span></strong> <o:p></o:p></span></p> <div align=center> <table class=MsoNormalTable border=1 cellpadding=0 style='mso-cellspacing:1.5pt; border:outset 1.5pt;mso-yfti-tbllook:1184;mso-padding-alt:0in 0in 0in 0in'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal style='margin:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_4" o:spid="_x0000_i1029" type="#_x0000_t75" style='width:57pt;height:137pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image007.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=59 height=139 src="HeleShawCells_files/image008.gif" v:shapes="Picture_x0020_4"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><span style="mso-spacerun:yes"></span></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1028" type="#_x0000_t75" style='width:85pt;height:135pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image009.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=87 height=137 src="HeleShawCells_files/image010.gif" v:shapes="Picture_x0020_5"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><span style="mso-spacerun:yes"></span></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_6" o:spid="_x0000_i1027" type="#_x0000_t75" style='width:61pt;height:135pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image011.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=63 height=137 src="HeleShawCells_files/image012.gif" v:shapes="Picture_x0020_6"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><span style="mso-spacerun:yes"></span></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_7" o:spid="_x0000_i1026" type="#_x0000_t75" style='width:41pt;height:135pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image013.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=43 height=137 src="HeleShawCells_files/image014.gif" v:shapes="Picture_x0020_7"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><o:p></o:p></span></p> </td> <td style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal style='margin:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_8" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:285pt;height:186pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="HeleShawCells_files/image015.gif" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=287 height=188 src="HeleShawCells_files/image016.gif" v:shapes="Picture_x0020_8"><![endif]></span><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'><o:p></o:p></span></p> </td> </tr> </table> </div> <p align=center style='margin-top:5.0pt;margin-right:.5in;margin-bottom:5.0pt; margin-left:.5in;text-align:center'><span class=GramE><em><span style='font-size:12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin; layout-grid-mode:line'>Characteristics of flames in <span class=SpellE>Hele</span>-Shaw cells.</span></em></span><em><span style='font-size:12.0pt;font-family:Arial; mso-bidi-theme-font:major-latin;layout-grid-mode:line'> Left: direct images (flames propagate from left to right.) Cell width (vertical direction in these images) 39 cm. Cell length (horizontal direction in these images) 60 cm, but images are cropped to show only flame front. Images from left to right: 7.2% CH<sub>4</sub> in air, horizontal propagation; 7.1% CH<sub>4</sub> in air, upward propagation; 7.1% CH<sub>4</sub> in air, downward propagation; 3.0% C<sub>3</sub>H<sub>8</sub> in air, horizontal propagation. Right: correlation of wrinkled front speed (S<sub>T</sub>/S<sub>L</sub>) with <span class=SpellE>Peclet</span> number. <o:p></o:p></span></em></p> <p style='margin-right:.5in;text-align:justify'><em><span style='font-size: 12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin;layout-grid-mode:line'><o:p>&nbsp;</o:p></span></em></p> <p style='margin-right:.5in;text-align:justify'><em><span style='font-size: 12.0pt;font-family:Arial;mso-bidi-theme-font:major-latin;layout-grid-mode:line'>Not proficient in English?<span style="mso-spacerun:yes"> </span><a href="http://www.autoteiledirekt.de/science/chemicke-predni-a-predni-sireni-plamene-v-hele-shaw-bunek"><span style='font-style:normal'>Try the Czech version</span></a>.</span></em></p> </div> </body> </html>